Scaling advantage of chaotic amplitude control for high-performance combinatorial optimization

The development of physical simulators, called Ising machines, that sample from low energy states of the Ising Hamiltonian has the potential to transform our ability to understand and control complex systems. However, most of the physical implementations of such machines have been based on a similar concept that is closely related to relaxational dynamics such as in simulated, mean-field, chaotic, and quantum annealing. Here we show that dynamics that includes a nonrelaxational component and is associated with a finite positive Gibbs entropy production rate can accelerate the sampling of low energy states compared to that of conventional methods. By implementing such dynamics on field programmable gate array, we show that the addition of nonrelaxational dynamics that we propose, called chaotic amplitude control, exhibits exponents of the scaling with problem size of the time to find optimal solutions and its variance that are smaller than those of relaxational schemes recently implemented on Ising machines.

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

In this study, we theoretically investigated a generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics in the context of non-equilibrium statistical mechanics. Using the ability of Loewner evolution, which enables encoding of non-equilibrium systems into equilibrium systems, we formulated the encoding mechanism of the SLE by Gibbs entropy-based information-theoretic approaches to discuss its advantages as a means to better describe non-equilibrium systems. After deriving entropy production and flux for the 2D trajectories of the generalized SLE curves, we reformulated the system’s entropic properties in terms of the Kullback–Leibler (KL) divergence. We demonstrate that this operation leads to alternative expressions of the Jarzynski equality and the second law of thermodynamics, which are consistent with the previously suggested theory of information thermodynamics. The irreversibility of the 2D trajectories is similarly discussed by decomposing the entropy into additive and non-additive parts. We numerically verified the non-equilibrium property of our model by simulating the long-time behavior of the entropic measure suggested by our formulation, referred to as the relative Loewner entropy.

Entropy minimization, convergence, and Gibbs ensembles (local and global)

We approach the subject of Statistical Mechanics from two different perspectives. In Part I we adopt the approach of Lanford and Martin-Lof. We examine the minimization of information entropy for measures on the phase space of bounded domains, subject to constraints that are averages of grand canonical distributions. We describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space. We also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy. We then show that, for fixed constraints, the parameters of the corresponding grand canonical distribution converge, as volume increases, to the corresponding parameters (derivatives, when they exist) of the thermodynamic limit entropy. In Part II, we use the Banach manifold structure on the space of finite positive measures to show that the critical points of the Gibbs entropy are grand canonical equilibria when the constraints are scalar, and local equilibria when the constraints are integrable functions. This provides a rigorous justification of the derivation of the Gibbs measures that appears often in literature.

Thermodynamics of structure-forming systems

Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations the approach is equivalent to the grand-canonical ensemble; for small systems we find significant deviations. We derive the detailed fluctuation theorem and Crooks’ work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed. We apply the results to several physical systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition.

Mass distributions of meteorites

ABSTRACT For at least five decades, the study of the mass distribution of meteorites has been carried out. This study aims to obtain the flux of material that comes to the Earth’s surface. For this, the observational data were modelled statistical distributions of the most varied types, derived from Gibbs entropy. However, it appears that the fragmentation process is probably complex in nature. Given this particularity, we model the mass distribution of meteorites using the q-exponential function, derived from Tsallis non-extensive statistical mechanics. This distribution is capable of modelling the entire observed spectrum of meteorite mass regardless of whether the specimens originate from the fragmentation of a single meteorite, belong to the same mineralogical group or type, or when are separated by collection sites on the Earth’s surface. We suggest that most meteorite samples are incomplete in certain mass ranges due to the action of the so-called gathering bias.

Reply to “What Is the Maximum Entropy Principle? Comments on ‘Statistical Theory on the Functional Form of Cloud Particle Size Distributions’”

We welcome the opportunity to correct the misunderstandings and misinterpretations contained in Yano’s comment that led him to incorrectly state that Wu and McFarquhar misunderstood the maximum entropy (MaxEnt) principle. As correctly stated by Yano, the principle itself does not suffer from the problem of a lack of invariance. But, as restated in this reply and in Wu and McFarquhar, the commonly used Shannon–Gibbs entropy does suffer from a lack of invariance for coordinate transform when applied in continuous cases, and this problem is resolved by the use of the relative entropy. Further, it is restated that the Wu and McFarquhar derivation of the PSD form using MaxEnt is more general than the formulation by Yano and allows more constraints with any functional relations to be applied. The derivation of Yano is nothing new but the representation of PSDs in other variables.